The Surprise Hanging Paradox

I read a version of this paradox many years ago, thought it was nonsense, but couldn’t work out why. Recently I read a different version and understood why it was a silly paradox. Here’s the usual formulation:

A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.

Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the "surprise hanging" can't be on Friday, as if he hasn't been hanged by Thursday, there is only one day left - and so it won't be a surprise if he's hanged on Friday. Since the judge's sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.

He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn't been hanged by Wednesday noon, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning, he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all. The next week, the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the above, was an utter surprise to him. Everything the judge said came true.

The mistake is to pay any attention to all that pseudo-logic. You’ve been told that one day next week, you’re going to be hung. And that it will be a surprise.

No it won’t.

You know it has to be one of the days next week, and at the moment you hear it you know each day has a 14% chance of being the day. As each day passes, the probability increases. That’s no basis for surprise. You can only be surprised if you think there is a 0% chance of it being the day.

On the day itself, your proper, downright cool reaction should be Wednesday, huh? Well, it had to be some day.

But what about all that nonsense-logic? The Judge’s ruling, is contradictory. Your hanging can't be a surprise if you know it's what awaits you. Real logic tells us that you can prove whatever you like from a contradictory statement. No wonder you can twist a bunch of noodle-logic to prove that you’re not going to be hung. The reason you can’t find anything wrong with the argument is that there isn’t anything wrong with the argument. The flaw is in the premises, and the argument distracts you from that.

Or you could say, it’s what happens when you treat a probabilistic concept like surprise as if it is a two-valued one. You can be a little surprised.